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Encyclopedia of Microtonal Music Theory

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jot

[Joe Monzo]

A term coined by Augustus de Morgan to designate the tiny interval which represents 100,000 * log10(2), thus obviating the need to calculate logarithms.

A jot is ~0.039863137 cent, or just under 1/25 of a cent. The exact value is 1/(25.085833...) = 1200/30103 of a cent.

A jot is calculated as the 30103rd root of 2, or 2(1/30103), with a ratio of approximately 1:1.000023026. It is an irrational number. This interval therefore divides the octave, which is assumed to have the ratio 2:1, into 30103 equal parts. Thus a jot represents one degree in 30103-EDO tuning.

One potential defect of using jots is that the familiar 12-EDO semitone does not come out with an integer number of jots, since 30103 does not divide evenly by 12. Thus, the 12-EDO semitone is ~2508.583333 = exactly 25087/12 jots.

The formula for calculating the jot-value of any ratio is: jots = log10(r) * [30103 / log10(2)] or jots = log2(r) * 30103, where r is the ratio.

Ellis states that John Curwen, a great reformer of a cappella singing who intended to acheive just-intonation in his performances, used jots for his measurements.

Note that Sauveur's "heptamerides" of 2(1/301) are related to the "jots", being simply a slightly less accurate rounding.

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jots calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = jots

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References

Ellis, Alexander. 1885.

Appendix XX, in his translation of:
Helmholtz, On the Sensations of Tone, p 437.
Dover reprint 1954.

De Morgan, Augustus. 1864.

"On the beats of imperfect consonances",
Transactions of the Cambridge Philosophical Society vol. 10, pp. 129-145.